Edexcel GCSE Maths: Calculus and Graphs Worked Solutions

Question

Here is a speed-time graph. Work out an estimate of the gradient of the graph at t = 2

Worked Solution

1. 1

   To estimate the gradient of a curve at a point, draw a tangent (a straight line that just touches the curve at that point) at t = 2.

2. 2

   Choose two well-separated points on the tangent line and read off their coordinates from the graph. For example, the tangent might pass through (0, a) and (3, b).

3. 3

   Calculate the gradient: gradient = (change in y) ÷ (change in x) = (b – a) / (3 – 0). The mark scheme gives an example of approximately 2.7 ÷ 3 ≈ 0.9.

4. 4

   Any answer in the range 0.7 to 1.1 is acceptable, depending on how accurately the tangent is drawn.

Question

The graph of y = f(x) is shown on the grid. On the grid, sketch the graph of y = f(–x) + 3.

Worked Solution

1. 1

   The transformation y = f(–x) reflects the graph of y = f(x) in the y-axis (every point (a, b) moves to (–a, b)).

2. 2

   The transformation y = f(–x) + 3 then shifts the reflected graph up by 3 units (every y-coordinate increases by 3).

3. 3

   Apply both transformations to the key points. For example, the point (–2, 3) on f(x) goes to (2, 3) under reflection, then to (2, 6) after the +3 shift. The point (2, –7) goes to (–2, –7) then to (–2, –4). The point (–6, –7) goes to (6, –7) then to (6, –4).

4. 4

   Sketch the new curve through the transformed key points: (2, 6), (–2, –4), (6, –4).

Question

Here is the graph of y = x2 – 2x – 2. Write down the coordinates of the turning point on the graph of y = x2 – 2x – 2

Worked Solution

1. 1

   The turning point of a parabola y = ax² + bx + c occurs at x = –b/(2a). Here a = 1, b = –2, so x = –(–2)/(2×1) = 2/2 = 1.

2. 2

   Find the y-coordinate by substituting x = 1: y = 1² – 2(1) – 2 = 1 – 2 – 2 = –3.

3. 3

   The turning point is (1, –3).